Tutorial : Indices Problem 1

Given the equation 2^{2x + 1} X 3^{x – 1} = 8^x X 3^2x. Show that 6^x = 2/3.

Taken from the Integrated Curriculum for Secondary Schools Additional Mathematics Form 4 2005 ed. page 76 Formative Exercise 5.1 Question 4

This question is quite simple as long as you remember the law of indices
The law of indices states that :
a^m X aⁿ = a^{m + n}
a^m ÷ aⁿ = a^{m – n}
(a^m)ⁿ = a^mn
(ab)ⁿ = aⁿbⁿ
(a/b)ⁿ = aⁿ/bⁿ

First, we create similarities between the terms.
For instance, we know that 8^x can be rewritten as 2^3x.
8^x = (2³)^x = 2^3x

We can then rewrite 2^{2x + 1} X 3^{x – 1} = 8^x X 3^2x as 2^{2x + 1} X 3^{x – 1} = 2^3x X 3^2x.

From 2^{2x + 1} X 3^{x – 1} = 2^3x X 3^2x, we shift the terms with twos (2) to the left, and the threes (3) to the right. By doing this, we are grouping them together to simplify or rather enable us to work out the problem.

We will end up with 2^{2x + 1} / 2^3x = 3^2x / 3^{x – 1}.

Recalling the law of indices,
2^{2x + 1} / 2^3x can be rewritten as 2^{2x + 1 – 3x} and simplified to 2^{1 – x}.
3^2x / 3^{x – 1} can be rewritten as 3^{2x – x + 1} and simplified to 3^{x + 1}.

Recalling the index law again,
We write 2^{1 – x} as 2 / 2^x.
We write 3^{x + 1} as 3^x X 3.

We then end up with 2 / 2^x = 3^x X 3.

We move 2^x to the right hand side and 3 to the left hand side.

We get 2 / 3 = 3^x X 2^x.
Simplifying the equation will give us 2/3 = 6^x.
We have then concluded and proved that 6^x = 2/3.

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